Inequalities

Introduction

The concept of greater than or more than is first understood  in  dealing with counts or unsigned whole numbers. Before the introduction of signs, that is negative and positive numbers, finite decimal expansions extend this idea of greater than or more than. A finite decimal expansion in particular counts the number of units, tenths, thousandths and so on that the number it represents can be divided into. Beyond this, students may be shown or pointed to the comparison of (unsigned) numbers with infinite decimal expansions.

With the introduction of positive and negative numbers and zero on say the real number line, the technical ideas of greater than differs from the common usage, or the introductory idea of comparison of by size or magnitude (apart from any signs that may be present).

For positive or non-negative numbers, saying a first number is more (positive) than a second number is equivalent to saying the first number is further away from zero or has a greater magnitude. So f or nonnegative numbers a and b, the phrase a is greater than b has either one and both of the foregoing two meanings. Each meaning implies the other. But in the comparison of non-negative with negative numbers and in the comparison of negative numbers, saying a first number is more positive than another is not equivalent to saying the first number has a greater magnitude.  

For comparison of pairs of real numbers which may be positive, zero or negative, we say a is greater than b and write a > b when and only when a is more positive than b, that is when a = b +c for some positive number or equivalently when a - b is positive. There-in lies a technical use of the phrase greater than which departs from and extends the greater in magnitude use in daily life of non-negative numbers or quantities. 

Due technical meaning I suggest for the rest of this lesson, we read a > b aloud as a is more positive than b. That convention during this lesson will help you understand and explain operations on inequalities. After this lesson, we go back to reading a > b aloud as a is greater than b but with the unspoken knowledge that we mean a is more positive than b. 

 The symbol > traditionally has been called the greater than sign. But its technical use is as follows. For real numbers a and b, we write a > b if and only if there is positive number c such that a = b + c. So a is c units more positive than b. Moreover we write a < b when and only when b > a.

Now we start again.

1. Comparison of Distance to Origin

Which number -5 or 3 has the greater distance to the origin? The answer is -5 as it is 5 units away from the origin while 3 is 3 units away.

The distance of a number to the origin is called is magnitude or absolute value.   It can be obtained by dropping the sign.  It can be obtained by changing the sign to positive. The distance of a number x to the origin (aka magnitude or absolute value) is given by the number x when x is positive or zero, and the negative of x, that is -x, when x is negative.

Distance to the origin of a coordinate system can be compared for points along a line, in a plane or in space. 

2.  The More Positive Sign

Consider the following questions:

• Which is more positive:  10 or 4?
• Which is more positive:    0 or 18?
• Which is more positive     -2 or 2?
• Which is more positive     - 4 or -1?

Clearly  

• 10 = 4 + 6 is 6 ones more positive than 4.
• 18 = 0 + 18 is 18 ones more positive than 0
• 2 = -2 + 4 is 4 ones more positive than 0
• -1 = -3 + 2 is 2 ones more positive than 0

Definition of More Positive Than Relation and Symbol >,

Given two real numbers a and b we write a >   b when and only when there is a positive number c such that a = b + c. So a is c ones more positive than b or equivalently, the number   c = a - b is positive. We call >, the more positive than sign.  

Inequality Keeping Theorem :  if a >   b and k >   0 then  ka >   kb 

Proof:  Assume a >   b. Then  a-b is positive. Therefore  k(a-b) is positive since a positive times a positive is a positive.  Hence ka - kb = k(a-b) is positive .  .So   ka - kb is positive and  ka >   kb. 

First Reciprocal Comparison Theorem :  if a >   b  and ab > 0 then  1/b >   1/a  Proof: Take k = 1/(ab).

Inequality Theorem :  if a >   b and k <   0 then  ka <   kb 

Proof:  Assume a >   b. Then  a-b is positive. Therefore  k(a-b) is negative since a negative times a positive is a positive.  Hence ka - kb = k(a-b) is negative  .  .So   kb - ka is positive and  ka <   kb. 

Second Reciprocal Comparision Theorem :  if a >   b  and ab < 0 then  1/b >   1/a  Proof: Take k = 1/(ab).

The proof is important. If you can follow it that is a sign that you are thinking algebraically and that you are starting to understand the shorthand role of letter and symbols in describing calculations (giving formulas or computation rules) and in describing equations and inequalities. 

Side Reversal Theorem:  If a >   b and  k < 0    then kb  >   ka

In words, if  a is more positive than b and k is a negative number then  kb  is more positive than ka. , 

Proof:  Assume a  >   b. Then  a-b is positive. Therefore  k(a-b)  negative since  negative  times a positive gives  a negative.  Hence ka - kb = k(a-b) is negative .  .So   kb - ka is positive and  kb >   k. 

3.  The More Negative Sign

Definition of More Negative Than

Given two real numbers a and b, we write a <   b when and only when b >   a. That is, when and only when b - a is positive or equivalently a - b is negative.  

By the Definition Theorem:  a >   b when and only when b <   a.

Proof: See definition.

---

Sign Keeping Theorem :  if a <   b and k >   0 then  ka >   kb 

If  a is more negative than b and k is a positive number then  ka  is more negative than kb. 

Proof:  Assume a <   b. Then  a-b is negative Therefore k positive implies k(a-b) is negative since a positive times a negative is a negative..  Hence ka - kb = k(a-b) is negative . So   ka - kb is positive and hence  ka > kb  or  ka <   k. 

Side Reversal Theorem:  If a <   b and k is negative  then ka >   kb or equivalently 

If  a is more negative than b and k is a negative number then  kb  is more positive than ka. 

Proof:  Assume a >   b. Then  a-b is positive. Therefore  k(a-b)  negative since a negative times a positive is a positive.  Hence ka - kb = k(a-b) is negative .  .So   kb - ka is positive and  kb >   k. 

Properties of the more positive and more negative relations

Theorem:  A  real number a is  positive when and only when  a >   0

Proof: Observe a = 0 + a. So if a is positive then  a = 0 + c where c= a is positive. Conversely, if a >   0 then a = 0 + c for some positive number c. But 0+ c = c. So a = c is positive. 

Theorem:  A  real number a is  negative when and only when  a <   0

Proof: Observe a

Sign Reversal Theorem:  If  a is more positive than b and k is a negative number then  kb  is more positive than ka. In symbols, if a >   b and 0  >   k  then kb >   ka or equivalently 

Proof:  Assume a >   b. Then  a-b is positive. Therefore  k(a-b)  negative since a negative times a positive is a positive.  Hence ka - kb = k(a-b) is negative .  .So   kb - ka is positive and  kb >   k. 

End with a Summary

In the Hypothetical World: To  separate the technical usage from  the common usage, the symbol > could be named  or renamed the more positive than symbols. This new name corresponds precisely to the technical meaning. With that convention, the phrase a greater than b could revert to the common usage and mean |a| > |b|, a comparison of magnitude. But we do not live in the hypothetical world, because of that we read

• a >  b aloud as a is greater than b while we mean a is more positive than b.
• |a| > |b| aloud as a is greater than b in magnitude.

Here we begin to see that symbols are not aligned with the common language and the symbols are worth a thousand words, or precisely, if they are not worth a thousand like a picture, they may require a thousand words to be understood. Ouch.

Finally, linear and Nonlinear Orderings are provided by more positive than and comparison of magnitude.

A number b is said to between two other numbers a and c if and only if there is a positive number q < 1 such that b=qa + (1-q)c.

Ordering of the real line by the relationship more positive than provides a linear ordering of the real line: for any three points a, b and c on the real line the relationships a < b <c imply that b is between a and c, that is, there is a positive number q < 1 such that b=qa + (1-q)c. See if you can explain why, rigourously. 

Ordering by magnitude provides a linear ordering of the positive numbers. For any three unsigned (positive) numbers a, b and c, the relationship a < b < c implies that b is between a and c. But for any three points a, b and c on the real line the relationship |a| < |b| < |c| does not imply that b is between a and c. So ordering by magnitude (or absolute value) of points on the whole real line is nonlinear.

Exercises:

Find the interval of x-values which satisfy the following inequalities:

1. 2x + 3 > 13
2. 5x + 2 <  7
3. 2x+ 3 < 13
4. 5x+7 > 7
5. 3x+3 > 5x -2
6. 7x - 8 < 10 x + 8

The end points of the interval may be fractions. 

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